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JOHN B. GAYLE, OLIVER 1. LACEY, and JAMES H. GARY Southern Experiment Station, Region V, Bureau of Mines, U. S. Department of the Interior, University, Ala.
Mixing of Solids Chi Square as a Criterion The mathematical treatment of the mixing problem based on chi square is applicable to a wider range of mixtures than any previously suggested, when sampling data are available as some form of counts. It is recomment 'ed for all such applications, regardless of the number or distribution a components I
F O R analytical purposes, mixing data for solids may be separated into two distinct categories:
To obtain illustrative data, a set. of experiments was carried out using the Bureau of Mines mixing wheel shown in Figure 1 (7). This mixer was designed for mixing samples of coal and coke; for the present study a straight-sided metal container was substituted for the usual glass bottle with a constricted section a t the top. As a source of readily distinguishable particles, quantities of 14- by 20-mesh roofing granules of different colors were obtained. These granules were similar in surface characteristics, shape, and density, were large enough to permit counting without the
1. Results expressed in terms of continuous variables such as weight, volume, and per cent when not used to represent counting data 2. Results expressed in terms of discrete variables-as some form of counting data The results of practical investigations usually fall into the first category (3, 4, 70). Investigators concerned with the theory of mixing have studied systems consisting of discrete particles of readily distinguishable substances and have used the standard deviation as a criterion of mixing for two-component or binomial distributions. Their results constitute an excellent foundation for future work on the mixing of discrete particles (2, 5,8,9). The present report describes a method for following the mixing process which is applicable not only to binomial distributions but to distributions consisting of any finite number of components distinguishable as discrete particles. Results of previous investigations have been critically reviewed by Weidenbaum and Bonilla (9) and by Lacey (5). In view of the successful application of the standard deviation to two-component mixtures, a similar approach using a dispersion measure applicable to multicomponent systems should represent a desirable improvement, and a survey of available measures indicated that the statistic, chi square (6), should be eminently suited to this purpose. This does not represent a fundamental departure from previous efforts, as for two-component systems chi square is related to the standard deviation when used as a measure of dispersion. I t is applicable to systems with any number of components, while the standard deviation is generally limited to two-component systems.
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use of a magnifier, and underwent no appreciable degradation during mixing. The first component was added to the mixing container, leveled, the second component added, and so on. The container was fastened to the wheel and rotated for a predetermined number of revolutions, after which it was removed and a metal top with openings aligned as guides for a specially designed sampling thief was substituted for the rubber top used during mixing. The thief was then inserted successively into each of the four guide holes distributed across the mixer top (Figure 2). This operation yielded
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Figure 1.
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A mixing wheel was used to obtain illustrative data VOL. 50, NO. 9
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GUIDE HOLE POSITION 2 I
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SAMPLING THIEF SAMPLING GUIDE
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SAMPLE POCKET LOCAT I ON
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WHITE BLUE SAMPLE POSITIONS ARE DESIGNATED I A , 18, 3 8 ETC. THE NUMBER REFERS TO SAMPLING GUIDE HOLE POSITION WHILE THE LETTER REFERS TO LOCATION OF SAMPLE POCKET IN THIEF.
Figure 2.
To obtain samples, a thief was substituted for the rubber top
a total of 12 point samples, each consisting of approximately 40 particles. The particles making up each sample were drawn out into separate columns of single particles with a spatula and the distribution of the components in the first 30 particles of each column was recorded. A typical set of counts is given in Table I, with chi square values calculated from Equation 1 :
Table I. Example of Computation of Chi Square Using Data for Run No. 15 Sample Position la lb IC 2a 2b 2c 3a 3b 3c 4a 4b 4c
Color Distributiona Red White Blue No. of Particles 3 11 10 10 11 6 17 16 13 8 7 8
9 1
5 5 5 4 2
4 6 10 7 2
18 18 15 15 14 20 11 10 11 12 16 20 Total
where
x2 = chi square 0 = observed number of particles of any given color in a sample E = corresponding expected number, based on average distribution of components in mixture Figure 3 (left) and Table I1 present data for six runs using equiweight mixtures of red, white, and blue particles. I n plotting the data, several points were shifted to avoid loss of detail while
X2
Table II. 6.10 5.21 0.28 0.28 0.68 3.33 10.84 7.78 2.84 3.38 0.68 4.44 45.84
Run
Chi Square after Different Numbers of Revolutions Revolutions
NO.
Expected Distribution
2
35
55
460.6 391.0 296.8 416.2 483.4 440.4
... 146.6
23
10-10-10 10-10-10 10-10-10 10-10-10 10-10-10 10-10-10
... 240.2
11
7 8 9 10 11 12
243.6 266.2 243.8 197.0
134.0 158.6 99.3 112.6
82.2 34.1 49.4 48.4 36.2 41.0
34.8 20.6 38.6 38.8 24.8 31.6
18.2 31.4 19.4 33.2 22.2 13.0
6 13 14 15 16 17
15-9-6 15-6-9 6-9-15 9-6-15 6-15-9 9-15-6
433.3 473.9 434.0 330.2 232.3 354.1
190.3 290.9 194.9 299.6 222.8 189.8
79.0 104.3 157.6 89.9 132.6 56.0
51.8 27.7 59.2 45.8 44.0 33.0
27.3 24.2 30.5 21.1 26.5 26.6
27.8 28.2 26.2 30.0 15.7 22.1
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a Expected color distribution for this run; 9 red, 6 white, and 15 blue particles.
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maintaining the same perpendicular distance from the curve. The values for chi square decrease with increasing numbers of revolutions, approaching a mean lower limit asymptotically. The zero revolution or maximum value for chi square is equal to the number of degrees of freedom multiplied by the number of particles counted for each sample, if it is assumed that samples are not taken at the interface. The number of degrees of freedom is defined as the number of items of data that can be supplied by a random process under the counting procedure used. The number of particles counted for each sample has a fixed value of 30; therefore, once the number of red and blue particles is recorded, the number of white particles is uniquely determined by difference. Thus, each sample contributes two degrees of freedom, the total for the 12 samples shown in Table I being 24. The chi square value for zero revolutions is therefore 30 x 24 or 720, as shown in Figure 3 (left). The mean lower limit for chi square, which is approached asymptotically with continued mixing, is the expected value for a random mixture and is equal to the number of degrees of freedom, 24 in this instance. One or both limits will be changed if the number of components, the number of samples, or the number of particles counted per sample is changed. Figure 3 (right) and Table I1 present similar data for a 15-9-6 distribution of particles: the order in which the components were added to the mixer was varied, all possible combinations being used. The smooth curve drawn through the points is identical with that shown in Figure 3 (left). Inspection of the data indicates no significant effects caused by changing either the proportions or order of charging of the mixture components. The fact that, for distributions of particles of the Type used, both the upper and lower limits for chi square are independent of the proportions in which the components are present represents a distinct improvement over treatments based on standard deviations for which both limits vary with the distribution of mixture components.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Figure 3.
Procedures for mixing, samplin$, and counting contribute little to scatter of data
Although a smooth curve drawn through the points appears to represent the average trends satisfactorily, the individual points exhibit considerable scatter. This is to be expected, as the reproducibility of chi square values is dependent on the number of degrees of freedom as shown by Equation 2.
where uxi = standard z2
N
deviation of observed values for chi square = average observed value for chi square = degrees of freedom
Because the expected value of chi square for a random mixture is equal to the number of degrees of freedom, this equation reduces to that given for a random mixture in most texts-Le., up = As N equals 24 for the data shown in Figure 3, Equation 2 corresponds to a theoretical standard deviation of 29y0 of the observed average values. Analysis of the individual results, treating all 12 runs shown in Figure 3 as replicates, yielded observed standard deviations ranging from 17 to 31 and averaging 24% of the average values. I n view of the limited number of results on which these computations were based, the agreement is considered satisfactory and it is concluded that the procedures used for mixing, sampling, and counting contribute little to the observed scatter of the data.
The main purpose of the experiments conducted to date has been to obtain illustrative data for the computation of chi square values rather than to obtain performance data or to study the mechanism of mixing. The results, however, also permit illustrations of the application of chi square values to these important aspects of mixing. With the system and mixer used, a random state is approached very quickly under conditions of the experiment. Although this appears to be a t variance with conclusions recently published by Barker, Mott, and Thomas (7), these workers used a range of particle sizes and
particle densities and the difference may be indicative of the influence of these variables. Further studies of the efficacy of this mixer appear desirable. For convenience in discussing the data with reference to the mechanism of mixing, a rate equation similar to those used by previous investigators has been formulated by combining the various chi square values to give “segregation indexes” (3)
where
S
= a numerical index which indi100
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Figure 4. Following an initial period of rapid mixing, the rate drops and the plot becomes linear VOL. 50, NO. 9
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EX PECTED DISTRIBUTION OF COMPONENTS 10 IO IO 15 9 6 10 20 0 20 IO 0 24 6 0 21 9 0 15 15 0
b
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REVOLUTIONS Figure 5. Rate of mixing depends mainly on mixer a i d particle characteristics
cates degree of segregation of the mixture xzo = observed chi square for any mixture xZr= expected chi square for random mixture xzS= expected chi square for segregated mixture This segregation index, which varies from unity to zero as mixing proceeds from complete segregation to complete randomization, may be used to establish a conventional rate equation relating the rate of change to the degree of segregation, as follows : -
ds = k(S)” dr
(4)
where dS = change in degree of segregation dr = change in number of revolutions of mixer k = rate constant S = measure of degree of segregation n = constant indicating order of the process Most workers have stated or assumed that mixing is a first-order process for which n = 1. Integration of Equation 4 for n = 1 indicates a linear relation between log (S x 100) and the number of revolutions. A plot of the data in accordance with this relation, however, does not yield a straight line over the entire range [Figure 4 (left)], but appears to indicate a n initial period of rapid mixing, after which the rate drops somewhat and the plot becomes apparently linear. Values for S equal to 2% or less
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are not plotted because of the large fluctuations caused by small variations in the observed values for chi square. Integration of Equation 4 for n = 2 indicates a linear relation between 1/S and the number of revolutions. A plot of the data in accordance with this relation also fails to yield a straight line over the entire range [see Figure 4 (right)], giving instead a n initial linear portion followed by a markedly nonlinear portion. Taken together, these graphs can be interpreted to indicate that for the particular mixer and system used, the mixing process takes place by a n initial second-order mechanism, followed by a slower first-order mechanism. This does not seem unreasonable in view of the nature of the mixing process. However, another possible interpretation is that the process is actually first order and the rapid mixing observed during the early stages is a t least partly the result of taking a small number of samples from fixed locations within the mixer. Because particles of one color are always in contact with those of another a t the interface between the several components, it is evident that complete segregation does not exist even a t zero revolutions of the mixer and this factor has been neglected in the present as well as most previous investigations. Although its importance can readily be assessed either by applying a correction to the zero revolution chi square value or by a more extensive sampling procedure involving large numbers of samples taken from random loca-
INDUSTRIAL AND ENGINEERING CHEMISTRY
tions in the mixer both before and after mixing, the magnitude of this effect is slight and the data now available do not warrant rejection of either mechanism. Further experimental studies are indicated. Use of segregation indexes as defined by Equation 3 to follow the mixing process is of advantage in comparing results obtained for mixtures consisting of different numbers of components. Thus, regardless of the number of components, when expressed in this manner the state of mixing varies from unity for a completely segregated system to zero for a random mixture. T o obtain illustrative data for a two-component system it is only necessary to combine certain of the results already presented and calculate new chi square values for systems consisting of, say, red, and blue plus white particles. Data obtained in this manner for two-component systems are compared in Figure 5 with those presented for the three-component systems. The results are consistent with a single curve, which suggests that under the conditions of the test, the rate of mixing depends mainly on mixer and particle characteristics and is substantially the same for mixtures of two and three components. Again this suggestion requires extensive experimental verification and work on this problem is continuing. Subsequent investigations are scheduled on the mechanism of mixing, mixer performance, mass transfer, and mixing in continuous mixers. The possibility of developing a method for starting with a system of segregated numerical digits and, by discrete and well-defined numerical operations, subjecting it to a randomizing process similar to that used in the study of random walks will also be considered. If such a n approach can be used to generate synthetic mixing data, it may prove useful in isolation and study of the different mechanisms of mixing. Literature Cited (1) Barker, J. E,, Mott, R. A., Thomas, W. C., Fuel 35, 493-9 (October 1956). ( 2 ) Blumberg, R., Maritz, J. S., Chem. Eng. Sci.2,240-6 (1953). (3) Clare, K. E., “Some Problems in Mixing Granular Materials Used in Road Construction,” Public Works and Municipal Services Congress, Nov. 15, 1954. (4) Gray, J. B., Chem. Eng. Progr. 53,25-32 (1957 \.
( 5 j Lacky, P. M. C., J. A$$. Chem. 4, 257-67 (1954). ( 6 ) Snedecor, G. W., “Statistical Methods,” 4th ed., Iowa State College Press, Ames, Iowa, 1950. (7) Stanton, F. M., Fieldner, A. C., Bur. Mines Tech. Paper 8, 9 (1913). (8) Weidenbaum, S. S., “Fundamental Study of Mixing of Particulate Solids,” doctoral thesis, Columbia University, 1953. ( 9 ) Weidenbaum, S. S., Bonilla, C. F., Chem. Eng. Progr. 51, 27J-36J (1955). (10) Work, L. T., Zbid.,50,476-9 (1954). RECEIVED for review October 30, 1957 ACCEPTEDMay 16,1958 I
